Show that all vectors in the vector field F(y,v)=(v,-y) are tangent to circles centered at the origin.
The hint I have is I can use slopes or dot product: I just need some help on how to use them.
Thanks in advance.
I'm gonna' give this a whirl alright. So the equation of a circle is:
$\displaystyle x^2+y^2=r^2$
differentiating:
$\displaystyle 2x+2yy'=0$
or:
$\displaystyle \frac{dy}{dx}=-\frac{x}{y}$
or parametrically:
$\displaystyle \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=-\frac{x}{y}$
and thus:
$\displaystyle \begin{aligned}
\frac{dy}{dt} &=-x \\
\frac{dx}{dt} &=y\end{aligned}
$
Also, you've got to get Mathematica or something else and just plot these to see globally what's happening. In Mathematica 7, you could use:
StreamPlot[{-x,y},{x,-3,3},{y,-3,3}]
ain't that simple or what?